Politeknik Dergisi
Cilt: 8 Sayı: 2 s. 123-130, 2005
Journal of Polytechnic
Vol: 8 No: 2 pp. 123-130, 2005
A SHEPWM Technique With Constant v/f for
Multilevel Inverters
*Firat
Servet TUNCER*, Yetkin TATAR**, Hanifi GÜLDEMİR*
University, Technical Education Faculty, Department of Electronic and Computer Science, Elazığ/TURKEY
**Firat University, Engineering Faculty, Department of Computer Engineering, Elazığ/TURKEY
ABSTRACT
This paper concern on a new application of selected harmonic elimination pulse width modulation technique (SHEPWM)
for multilevel inverters. In this paper, first, the switching angles are calculated using constrained optimization technique. With
these switching angles both the fundamental harmonic can be controlled and the selected harmonics can be eliminated. Then,
using these calculated switching angles, a set of equation is formed which calculates the switching angles with respect to
modulation index. Using this technique three-phase voltage has been obtained from a five-level cascade inverter. This voltage is
applied to an induction motor. The dynamic behaviour of the induction motor has been examined for constant v/f operation and
the simulation results have been given.
Keywords: Multilevel inverter, SHEPWM technique, optimization theory, induction motor, v/f operation.
Çok Seviyeli İnverterler için Sabit v/f Özellikli Bir
SHEPWM Tekniği
ÖZET
Bu makale, çok seviyeli inverterler için seçilmiş harmonikleri yok eden darbe genişlik modülasyon (SHEPWM)
tekniğinin yeni bir uygulamasını açıklamaktadır. Makalede ilk olarak, sınırlayıcılı optimizasyon tekniği kullanılarak anahtarlama
açıları hesaplanır. Bu anahtarlama açıları ile hem temel harmonik kontrol edilebilir ve hem de seçilen harmonikler yok edilebilir.
Sonra, bu hesaplanmış anahtarlama açıları kullanılarak; modülasyon indeksi değişkenine göre anahtarlama açılarını hesaplayan
denklem kümesi elde edilmiştir. Bu denklemler kullanılarak 5-seviyeli kaskad inverterden üç-fazlı gerilim Elde edilmiştir. Bu
gerilim bir asenkron motora uygulanmıştır. Sabit v/f çalışma için asenkron motorun dinamik davranışı incelenmiş ve
simülasyon sonuçları verilmiştir.
Anahtar Kelimeler: Çok seviyeli inverter, Seçilmiş harmonikleri yok eden darbe genişlik modülasyon tekniği,
Optimizasyon teorisi, asenkron motor, v/f çalışma.
1. INTRODUCTION
Multilevel inverters have become an effective
and practical solution for increasing power and reducing
harmonics of ac waveforms. The main advantages of
multilevel PWM inverters are the following: 1) The
series connection allows high voltage without
increasing voltage stress on switches. 2) Multilevel
waveforms reduce the dv/dt at the output of an inverter.
3) At the same switching frequency, a multilevel
inverter can achieve lower harmonic distortion due to
more levels of the output waveform in comparison to a
two level inverter (1). Various multilevel inverter
topologies have been proposed and implemented (1-7).
There are three reported basic topologies of multilevel
inverters: Diode-clamped, Capacitor-clamped, and
Cascade multilevel inverters (7,9).
The concept of diode-clamped multilevel
inverters was first introduced by Nabae in 1981. The
general structure of the multilevel inverter is to
synthesize a sinusoidal voltage from several levels of
voltages, typically obtained from capacitor voltage
sources. Unfortunately this structure is quite limited not
only due to voltage unbalance problems but also due to
voltage clamping requirements (2).
In order to eliminate the clamping diodes the
capacitor-clamped topology has been developed and
used in many applications. The voltage level of the
capacitor-clamped multilevel inverter is similar to that
of the diode-clamped multilevel inverter. However, the
advantage of the capacitor-clamped inverter is to have
two or more switching combinations at the mid levels of
the output voltage (7).
The cascade multilevel inverter topology which
is formed by series connections of one phase bridge
type inverters (H-bridge) is simpler than the other two
types of topology and packaging is possible. The
number of output voltage levels can be easily adjusted
by adding or removing the full bridge cells. The output
voltage of cascade multilevel inverters is determined by
the synthesation of the voltage of the isolated dc
sources.
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Servet TUNCER, Yetkin TATAR, Hanifi GÜLDEMİR / POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
If the three multilevel inverters mentioned above
are compared each other all require the same number of
main switches per phase leg. The diode-clamped
multilevel inverters require extra clamping diodes, the
capacitor-clamped topology require extra balancing
capacitor and the cascade multilevel inverter topology
needs additional isolated dc sources.
Each phase of the multilevel inverters is formed
by series or parallel connection of the switching
devices. This topology decreases the power ratings of
the switching device and increase the output voltage
and output power.
It is generally accepted that the performance of
an inverter with any switching strategies, can be related
to the harmonic contents of its output voltage. As the
number of input voltage levels increases the output
waveform approaches the sinusoidal wave with
minimum harmonic distortion (8). Power electronic
researchers have always studied many novel control
techniques to reduce harmonics in such waveforms. In
multilevel topology several well-known modulation
techniques are used. These modulation techniques are
Subharmonic Pulse Width Modulation (SPWM), Space
Vector Pulse Width Modulation (SVPWM), and
Selected Harmonic Elimination Pulse Width
Modulation (SHEPWM) (3,6,9,10).
SPWM strategies for multilevel inverters employ
extensions of carrier based techniques used for two
level inverters. The control principle of the SPWM
method is to use several triangular carrier signals with
only one modulation wave per phase. It has been shown
that the spectral performance of a multilevel waveform
can be significantly improved by employing alternative
dispositions and phase shifts in the carrier signals,
however, the side band harmonics occur at the carrier
frequency. As the carrier frequency is much higher than
the fundamental, those harmonics in the output voltage
are not so important and can be eliminated using filter
circuits (3,13)
In the SVPWM technique; any three-phase
quantities e.g. three-phase voltages and currents can be
represented by a space vector in a d-q plane via Park’s
transformation. The vector starts from the origin and
ends at the certain point so that the length and the phase
angle of the vector together represent the instantaneous
values of the particular three-phase quantities [8]. In the
SVPWM technique, the calculation of dwell-time and
selection of sectors are similar to that of the two level
inverters. The hardware implementation of the
technique is simple and the current ripples are very low.
Therefore SVPWM technique is suitable for high
power/voltage applications (9).
proper off-line calculations. The results are then either
directly stored in look-up tables or interpolated by
simple functions for real time operation. The SHEPWM
based methods can theoretically provide the highest
quality output among all the PWM methods. The
disadvantage of this technique is that if the number of
switching angles are increased then the look up table
requires much memory space.
In this paper, a new application of the SHEPWM
technique used in the multilevel inverters is introduced.
The simple algebraic equations which calculates the
switching angles according to given modulation index
are formed. These algebraic equations use precalculated switching angles according to predefined
criteria’s. As these equations can be easily solved by the
microprocessors or DSP’s the use of look-up table is,
thus, eliminated.
Using this technique, the three-phase voltage are
obtained from a five-level cascade inverter and applied
to an induction motor. The dynamic behaviour of the
induction motor is examined and the simulation results
are given.
2. MULTILEVEL SHEPWM TECHNIQUE
SHEPWM technique is first introduced by Patel
and Hoft and it is an effective technique for the
elimination of low order harmonics in two and threelevel inverters (11). This technique is further extended
to be used in multilevel inverters.
Figure 1 shows three-phase structure of a fivelevel cascade inverter. As can be seen in the figure, each
phase consist of two series connected H-bridge inverters
H1 and H2. Each H bridge cell is fed by an isolated
separate dc source. The output phase voltage of the
inverter is the sum of the output voltage of these cells.
The switching angles for each cell of the fivelevel cascade inverter are calculated by using
SHEPWM technique for only a quarter of a period.
Figure 2 illustrates the general quarter wave symmetric
five-level SHEPWM switching pattern. α i and β i ’s
are the switching angles of the cells H1 and H2
respectively. Each cell of such a single-phase inverter
switches m times per quarter cycle. Owing to the
symmetries in the PWM waveforms, only the odd
harmonics exist.
Selected harmonic elimination PWM technique
is introduced by Patel and Hoft (11). The idea of the
technique is that the basic square-wave output is “
chopped ” a number of times, which are obtained by
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A SHEPWM TECHNIQUE WITH CONSTANT V/F FOR MULTILEVEL INVERTERS …/ POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
S1
S2
V
S3
S4
S1'
S2'
S3'
S4'
a
V1
S1
S2
b
V
S1
S2
V
S3
S4
S3
S4
S1'
S2'
S1'
S2'
S3'
S4'
S3'
S4'
It is very difficult to solve such a set of equations
numerically due to the convergence problem and it also
requires considerable computation. For such problems,
in order to overcome the computational problems a
constrained optimization approach has been proposed
(4,6). For the full solution of this optimization scheme,
the cost function and the constraints for α and β
switching angles can be written as,
c
H1
V
V2
V
V
Cost function:
H2
F = ( V1 − 2 M ) 2 + V5 2 + V7 2 + ......
n
Figure 1. Three-phase structure of a five-level cascade
inverter.
H1
Constraints:
V
-V
π/2
α1
α2 ....α m
π
0 < α 1 < α 2 < ..... < α m < π / 2
2π
0 < β1 < β 2 < ..... < β m < π / 2
(a)
H2
β1
-V
π/2
β 2 ....β m
Van
π
2π
(b)
2V
V
π/2
π
(3)
Where F is the cost function, M is the
modulation index, V1 is the amplitude of the
fundamental, and, V5, V7,... are the amplitudes of the 5th
and 7th selected harmonics which are going to be
eliminated.
V
2π
-V
-2V
(2)
(c)
Figure 2. Waveform of five-level SHEPWM.
(a) Output voltage of H1 bridge.
(b) Output voltage of H2 bridge.
(c) Phase voltage.
The amplitude of the nth harmonic of the inverter
phase voltage can be written from the sum of the output
voltages of the cells H1 and H2 as (6).
4V
(cos nα 1 − cos nα 2 + ....... ± cos nα m + (1)
nπ
cos nβ 1 − cos nβ 2 + ....... ± cos nβ m )
Vn =
Matlab/Optimization toolbox is a very useful
software package for the solution of multi-variable
constrained optimization problems. In this paper,
“fmincon” function is used in the algorithm for the
determination of α and β values. The figure 3 shows
the variation of switching angles as a function of
modulation index. These results have been obtained by
taking 3 switching angles for each bridge cell and using
equations (1-3) the 5th, 7th, 11th and 13th harmonics are
eliminated in the range between 0.05-1.15 of
modulation index. The figure 4 shows the results for the
elimination of 5th, 7th, 11th, 13th, 17th, 19th, 23rd and 25th
harmonics in the same range of modulation index. This
time 5 switching angles are used.
As there are no triplen harmonics in three-phase
star connected isolated-neutral systems, there is no need
to extend the algorithm to eliminate the triplen
harmonics in the solution of the nonlinear equation set
(14). In general, the most significant low-frequency
harmonics are chosen for elimination by properly
selecting angles among different level inverters, and
high frequency harmonic components can be readily
removed by using additional filter circuits.
Where n is the order of the harmonics, and V is
the input dc voltages of the cells H1 and H2 respectively.
Theoretically, 2m-2 odd harmonics can be
eliminated by solving 2m-1 nonlinear equation. This is
achieved by equating the amplitudes of the selected
harmonics to zero and setting the fundamental to a
desired value.
125
Servet TUNCER, Yetkin TATAR, Hanifi GÜLDEMİR / POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
switching angles are determined and stored in the lookup table. In order to find the switching angles for any
modulation index the step must be kept very small. This
makes look-up table to occupy much memory space
and making the algorithm very long.
90
80
Switching Angles (Deg.)
70
60
50
40
30
H1
20
H2
10
0
0.2
0.4
0.6
0.8
1
Modulation Index
Figure 3. Switching angles as a function of modulation index
(m=3 ).
90
80
In the analysis, the number of region is kept the
same both for m=3 and m=5. If the number of region is
decreased then the order of polynomial should be
increased. However, this would increase computation
time. Equations (4) show the variation of switching
angles as a function of M in the second region for m=3.
70
Switching Angles (Deg.)
In this study, a new technique is used for the
computation of the switching angle during real-time
operation. This technique avoids the use of look-up
tables for the computation of the switching angles. The
technique is based on the simple functions, which
represent the off-line calculated solution trajectories and
obtained by a suitable curve fitting technique. Thus, the
output of the optimal SHEPWM switching strategy is
represented as a set of curves which define the
switching angles for any given modulation index. The
modulation index between the values of 0.05 and 1.15 is
divided into five regions. Third order polynomial curve
fitting is applied to every region. “Polyval” function of
the Matlab has been used for the construction of the
polynomial functions. Figure 5 shows these regions. As
the switching angles vary approximately linear with low
values of modulation index, the first region is kept
larger than the others.
60
50
α1 = 431.7357 M3 - 724.9738 M2 + 360.8806 M + 0.0001
40
α2 = 263.4361 M3 - 614.4191 M2 + 401.2735 M + 0.0002
30
α3 = 196.1671 M3 - 569.5637 M2 + 409.3648 M + 0.0001
20
H1
β1 = 211.5309 M3 - 399.5012 M2 + 229.5341 M + 0.0002 (4)
10
H2
β2 = 291.2417 M3 - 530.5984 M2 + 287.3848 M + 0.0003
β3 = 559.3158 M3 - 977.3094 M2 + 501.9763 M + 0.0001
0
0.2
0.4
0.6
0.8
1
Modulation Index
Figure 4. Switching angles as a function of modulation index
(m=5 ).
The switching angles obtained from 3rd order
polynomials and that obtained from optimization
technique are very close to each other. This is shown in
Tables 1 and 2 for M=0.6.
Table 1. Comparison of switching angles for m=3 and M=0.6
3. EQUATIONS FOR THE DETERMINATION
OF SWITCHING ANGLES
Due to the high complexity, the determination of
switching angles on-line during real-time operation is
considered to be impractical. Hence, firstly the
equations are solved off-line. The results are then stored
in look-up tables for real-time operations. The switching
angles corresponding to any modulation index can be
easily determined using this look up table. The look-up
table stores the switching angles, which correspond to
modulation index. For this, the modulation index
increased with steps starting from an initial value and
for every step of modulation index, the corresponding
126
Switching Calculated
Angles value (Deg.)
The value
obtained by
curve fitting
(Deg.)
Error
(%)
α1
48.7536
48.7928
0.080
α2
76.2644
76.4756
0.276
α3
82.8297
82.9481
0.143
β1
39.4305
39.5909
0.406
β2
44.1449
44.3240
0.405
A SHEPWM TECHNIQUE WITH CONSTANT V/F FOR MULTILEVEL INVERTERS …/ POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
β3
70.0633
70.1667
0.147
Input
frequency
Table 2. Comparison of switching angles for m=5 and M=0.6
Switchin
g Angles
Calculated
value (Deg.)
The value
obtained by
curve fitting
(Deg.)
Error
(%)
Compute M
α1
38.7802
38.9778
0.509
Find the region and
compute switching angles for
that region
α2
42.9010
43.2317
0.770
α3
59.8143
60.0605
0.411
α4
80.2979
80.6724
0.466
α5
84.5050
84.7412
0.279
β1
47.7368
47.9793
0.507
β2
58.1824
58.7566
0.986
β3
63.2534
63.9352
1.000
β4
66.3079
66.9324
0.941
β5
74.2243
74.5528
0.442
Use this switching angles
produce three phase signals
with quarter wave symetry
No
Yes
Figure 6. Flowchart for PWM waveform
90
4. ANALYSIS OF THREE-PHASE AC MOTOR
FED BY FIVE-LEVEL CASCADE INVERTER
80
The simulation of an ac motor fed by a five-level
cascade inverter has been carried out using the
SHEPWM waveform as explained in the previous
sections. For the simulation, the equations for the
switching angles for each region shown in figure 5 in
the range of 0.05-1.15 modulation index have been
constructed. For any frequency, the one of the region in
figure 5, which corresponds to the value of M
maintaining constant v/f, is determined. The switching
angles are calculated from the algebraic equations of the
determined region. Using these switching angles, threephase PWM waveforms with quarter wave symmetry is
obtained. These three-phase voltage is applied to d-q
model of an induction motor. Matlab/Power System
Blockset has been used for the simulation and the block
diagram is given in figure 7.
70
Switching Angles (Deg.)
is frequency
changed ?
60
50
40
30
20
10
I
II
III IV V
0
0.2
0.4
0.6
0.8
Modulation Index
1
Figure 5. Constructed regions for m=3
Using the equations for every region shown in
figure 5, both the amplitude and the frequency of the
inverter output voltage can be controlled. It has been
chosen that the modulation index M=1 to be occur at 50
Hz fundamental frequency. Thus the frequency range of
2.5-57.5Hz is obtained which corresponds to 0.05-1.15
modulation index range. The accuracy of the on-line
generated switching angles depend on the proximity of
functions and could be increased using higher order
polynomials.
A s-function has been used for the calculation of
switching angles. A time varying input frequency is
used during the acceleration to maintain constant V/f
ratio. The PWM waveforms for each H-bridge are then
constructed according to the flowchart given in figure 6.
5 Nm load torque has been applied to the motor during
the simulation. The other parameters are tabulated in
Table 3.
Table 3. Motor parameters
In this technique only the coefficients of the
polynomial have to be saved in the memory. The flow
chart of the algorithm is shown in figure 6.
127
Nominal Power
Line-line voltage
Frequency, f
Stator resistance, Rs
Rotor resistance, Rr
Stator inductance, Ls
Rotor inductance, Lr
Mutual inductance, Lm
3 Hp
220V
50Hz
0.435 Ω
0.816 Ω
2mH
2mH
69.31mH
Servet TUNCER, Yetkin TATAR, Hanifi GÜLDEMİR / POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
0.089 kgm2
2
Fcn
f(u)
C1
C2
B1
B2
time
A1
A2
s-function
H-Bridges
(leg_A)
H-Bridges
(leg_B)
C
time
a
Asynchronous
Machine
m_SI Tm
Clock
B
t
A
The simulation results have been presented in
figure 8-11. The simulations have been done for m=3,
m=5 and for f=25 and f=50 Hz fundamental
frequencies. As can be seen from the simulation results,
both selected harmonics are eliminated and a variable
amplitude and frequency ac voltages are obtained.
freq.
b
c
Inertia, j
Pairs of poles, p
H-Bridges
(leg_C)
5
is_abc
m wm
Te
Figure 7. Block Diagram of the system.
128
A SHEPWM TECHNIQUE WITH CONSTANT V/F FOR MULTILEVEL INVERTERS …/ POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
200
Line voltage (V)
Line voltage (V)
200
100
0
-100
-200
0.42
0.43
0.44
0.45
0.46 0.47
Time (sec.)
0.48
0.49
-100
80
60
40
20
0
250
500
750
1000 1250
Frequency (Hz)
1500
0.475 0.48 0.485
Time (sec.)
0.49
0.495
500
750 1000 1250
Frequency (Hz)
1500
1750 2000
150
100
50
0
1750 2000
0
250
Speed (rad./sec.), Torque (Nm) Line to line voltage (V)
(a)
200
0
-200
0.05
0.1 0.15 0.2
100
0.25 0.3
Time (sec.)
0.35
0.4 0.45 0.5
50
0
-50
0
0.05 0.1
0.15 0.2
50
0.25 0.3 0.35 0.4
Time (sec.)
0.45 0.5
400
200
0
-200
-400
0.05 0.1
0.15 0.2
0.25 0.3
Time (sec.)
0.35 0.4
0.45 0.5
0
0.05 0.1
0.15 0.2
0.25 0.3
Time (sec.)
0.35 0.4
0.45 0.5
0
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
100
0
-100
100
0
-50
0
0
200
Stator current (A)
Stator current (A) Speed (rad./sec.), Torque (N.m) Line to line voltage (V)
(a)
0
0
-100
0.05
0.1
0.15 0.2
0.25 0.3
0.35
0.4 0.45 0.5
Time (sec.)
Time (sec.)
(b)
Stator Current (A)
Stator Current (A)
(b)
15
10
5
0
-5
-10
-15
0.42 0.43
0.44
0.45
0.46 0.47
Time (sec.)
0.48
0.49
0.47
0.475 0.48 0.485
Time (sec.)
0.49
0.495
0.5
500
750
1000 1250
Frequency (Hz)
1500
1750
2000
10
Amplitude (A)
Amplitude (A)
15
10
5
0
-5
-10
-15
0.46 0.465
0.5
10
8
6
4
2
0
0.5
0.47
200
Amplitude (V)
Amplitude (V)
0
-200
0.46 0.465
0.5
100
0
100
0
250
500
750
1000 1250 1500
Frequency (Hz)
1750
8
6
4
2
0
2000
0
250
(c)
(c)
Figure 8.Waveforms for m=3 and f=25Hz.
Figure 9.Waveforms for m=3 and f=50Hz.
129
Line to line voltage (V)
Line to line voltage (V)
Servet TUNCER, Yetkin TATAR, Hanifi GÜLDEMİR / POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
200
0
-200
0.42
0.43
0.44
0.45
0.46 0.47
Time (sec.)
0.48
0.49
0.5
400
200
0
-200
-400
0.46 0.465
0.47
0.475 0.48 0.485 0.49
Time (sec.)
0.495 0.5
Amplitude (V)
Amplitude (V)
200
150
100
50
0
0
250
500
1000
750
Frequency (Hz)
1250
300
200
100
0
1500
0
250
500
750 1000 1250
Frequency (Hz)
1500
1750 2000
Figure 11.Waveforms for m=5 and f=50Hz.
Figure 10.Waveforms for m=5 and f=25Hz.
5. CONCLUSION
In this paper, a new application of SHEPWM
technique which both eliminates the selected harmonics
and controls the fundamental component of the line
voltage has been studied. Three-phase voltage has been
obtained from a five-level cascade inverter using the
switching angles determined from algebraic equations.
The three- phase voltage is applied to an induction
motor and dynamic behaviour of the induction motor
has been examined under v/f control.
In the proposed SHEPWM application, there is
no need to save all the switching angles which
correspond to modulation index values as in the
conventional SHEPWM technique. Here, only the
coefficients of the algebraic equations are saved in the
memory. Hence, less space has been
used in the memory, which makes it suitable to
implement practically using microprocessors and
DSP’s.
The simulation results show that using the
proposed technique of the voltage can be controlled in a
wide range. If the number of selected harmonics which
are going to be eliminated from the line voltage are
increased either the level of the inverter or the number
of switching angle in a quarter period needs to be
increased.
6. REFERENCES
1. Li L.., Czarkowski D., Liu Y., and Pillay P.,:Multilevel
Space Vector PWM Technique Based on Phase-Shift
Harmonic Suppression. Applied Power Electronics
Conference and Exposition, 2000. APEC 2000. Fifteenth
Annual IEEE, Vol.1, pp. 535-541, February 2000
2. Nabae A., Takahashi I., Akagi H., A new Neutral-PointClamped PWM Inverter, IEEE Trans. Industry
Applications, Vol. 19, No 6, pp. 1057-1069, Nov/Dec.
1983.
3. Sirisukpraset S., Lai J.S., Liu T.H., Optimum Harmonic
Reduction with a Wide Range of Modulation Indexes for
Multilevel Converters. Annual Power Electronics
Seminar, September 19-21, 1999.
4. Lund R., Manjrekar M.D., Steimer P. and Lipo T.A.
Control Strategies for a Hybrid Seven-Level Inverter,
European Power Electronics Conference, pp. 1-10, Sept.
1999, Lausanne Switzerland.
5. Tolbert L.M., Peng P.Z, Habetler T.G., Multilevel
Converters for Large Electric Drives, IEEE Transactions
on Industry Applications, Vol. 35, No. 1, pp. 36-44,
Nov/Dec. 1983.
6. Li L.., Czarkowski D., Liu Y., and Pillay P., Multilevel
Selective Harmonic Elimination PWM Technique in
Series-Connected Voltage Inverters, IEEE Trans. Ind.
App. Vol. 36, No. 1, pp. 160-170, January/February 2000.
7. Lai J.S., and Peng F.Z., Multilevel Converters - A New
Breed of Power Converters, IEEE Transactions on
Industry Applications, pp.509-517, May/June 1996.
8. Zhang H., Jouanne A.V., and Wallace A, Multilevel
Inverter Modulation Schemes to Eliminate Common
Mode Voltages, IEEE Industry Applications Conference,
Thirty-Third IAS Annual Meeting. Vol.1, pp. 752 –758,
1998.
9. Celanovic N., Space Vector Modulation and Control of
Multilevel Converters, Doctor of Philosophy, Faculty of
the Virginia Polytechnic Institute and State University, 20
September 2000.
10. Quin J., Lipo T.A., Switching Angles and DC Link
Voltages Optimization for Multilevel Cascade Inverters.
Electric Machines and Power Systems Conference, 1999.
11. Patel H.S. and Hoft R.G, Generalized technique of
harmonic elimination and voltage control in thyristor
inverters, Part I: harmonic elimination, IEEE Trans. Ind.
App., Vol. IA-9, pp. 310-317, May/June 1973.
12. Sun J., Beineke S., Optimal PWM Based on Real-Time
Solution of Harmonic Elimination Equations, IEEE
Transactions on Power Electronics, Vol.11, No.4, pp.612621, July 1996.
130
A SHEPWM TECHNIQUE WITH CONSTANT V/F FOR MULTILEVEL INVERTERS …/ POLİTEKNİK DERGİSİ, CİLT 8, SAYI 2, 2005
13. Carrara G., Gardella S., Marchesoni M., Salutari R.,
Sciutto G., A New Multilevel PWM Method: A
Theoretical Analysis, IEEE Transactions on Power
Electronics, Vol.7, No.3, pp. 497-505, July 1992.
14. Tuncer S., Tatar Y., Harmonic Optimization Method for
Cascade Multilevel Inverters, 2nd FAE International
Symposium, pp. 457-460, November 2002.
131